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ListComprehensiveGcd

The ListComprehensiveGcd function will compute a complete case discussion of the greatest common divisor of a list of parametric univariate polynomials.

Calling Sequence

ListComprehensiveGcd(lp, v, R, options)
ListComprehensiveGcd(lp, v, rs, R, options)
ListComprehensiveGcd(lp, v, cs, R, options)
ListComprehensiveGcd(lp, v, F, R, options)
ListComprehensiveGcd(lp, v, F, H, R, options)

Input

Variable Description
lp A list of parametric, univariate polynomial in v
v The variable that p1 and p2 are polynomials in
rs Regular system
cs Constructible set
F List of polynomials over R representing equality constraints on the parameters
H List of polynomials over R representing inequation constraints on the parameters
R Polynomial ring

Options

Option name Default value Description
'outputType' 'CS' When set to 'CS' or 'ConstructibleSet' the output will contain constructible sets, when set to 'RS' or 'RegularSystem', the output will contain regular systems.

Output

A sequence gcdList, cs_zero where gcdList is a list with elements of the form:

Output structure Options
[g, rs] 'outputType' is 'RegularSystem' or 'RS'
[g, cs] 'outputType' is 'ConstructibleSet' or 'CS'
[g, cof_p1, cof_p2, rs] 'outputType' is 'RegularSystem' or 'RS' and 'cofactors' = true

Where g_i is the gcd of p1 and p2 for all values in the zero set of cs or rs, and cof_p1 and cof_p2 are the cofactors of p1 and p2 respectively: cof_p1 = p1/g, cof_p2 = p2/g. cs_zero is the constructible set where both p1 and p2 vanish for all values in its zero set. The set of constructible sets returned (including cs_zero) form a partition of the input constructible set.

Assumptions

  1. v is the greatest variable of R

Example

with(RegularChains):
with(ConstructibleSetTools):
with(ParametricMatrixTools):

# Set up the polynomials with variable order x > a
R := PolynomialRing([x, a]):
lp := [x+1, (x+a)*(x-a), x-a]:

# Compute the gcd
g, cs_zero := ListComprehensiveGcd(lp, x, [], R);

# Print the result
IsEmpty(cs_zero, R);
    true
Display(g, R);
    [[1, a+1 <> 0], [x-a, a+1 = 0]]